PRICING CREDIT RISK DERIVATIVES
PHILIPP J. SCHONBUCHER¨
Abstract. In this paper the pricing of several credit risk derivatives is discussed in an intensity-based framework with both risk-free and defaultable interest rates stochas- tic and possibly correlated. Credit spread puts and exchange options serve as main examples. The differences to standard interest rate derivatives are analysed. Where possible closed-form solutions are given.
1. Introduction
Credit derivatives are derivative securities whose payoff depends on the credit quality of a certain issuer. This credit quality can be measured by the credit rating of the issuer or by the yield spread of his bonds over the yield of a comparable default-free bond. In this paper we will concentrate on the latter case. Credit Risk Derivatives (CRD) have enjoyed much attention and publicity in the deriva- tives world in recent years. They have been hailed as the major new risk management tool for managing credit exposures, as the most important new market in fixed in- come derivatives and as a breakthrough in loan risk management. This praise is at least to some part justified: Credit risk derivatives can make large and important risks tradeable. They form an important step towards market completion and efficient risk allocation, they can help bridge the traditional market segmentation between corporate loan and bond markets. Despite their large potential practical importance, there have been very few works specifically on the pricing and hedging of credit risk derivatives. The majority of papers is concerned with the pricing of defaultable bonds which is a necessary prerequisite for credit derivatives pricing. The leading recovery model in this paper is the fractional recovery / multiple defaults model (see Duffie and Singelton [7, 8] and also Sch¨onbucher [17, 18]). Applying this model to credit derivatives will yield important insights into the subtler points of the model used. The choice of model is not essential to the main point of the paper which is to show the methodology that has to be used in an
Date: July 1997, this version January 1998. Financial support by the DFG, SFB 303 at Bonn University and by the DAAD are gratefully acknowledged. I would like to thank Erik Schl¨ogland David Lando for helpful discussions. Comments and suggestions to this paper are very welcome. This is an early draft. All errors are of course my own. Author’s address: Department of Statistics, Faculty of Economics, University of Bonn, Adenauerallee 24-42, 53113 Bonn, Germany. email: schonbuc@addi.finasto.uni-bonn.de. 1 2 PHILIPP J. SCHONBUCHER¨ intensity-based credit spread model. It is easy to transfer most of the results to other intensity-based models like e.g. the models by Jarrow and Turnbull [12], Lando [14, 13], Madan and Unal [16] or Artzner and Delbaen [1], in fact it is demonstrated that under some circumstances it may be more appropriate to model the recoveries following one of these models. In the following section this model will be recapitulated, followed by a presentation of the credit derivatives that will be priced. As the vast majority of credit derivatives are OTC derivatives, no standard specification has evolved for second generation derivatives yet. Therefore we chose an array of typical and/or natural specifications to exemplify typical cases, including default swaps, straight put options on defaultable bonds, credit spread puts with different specifications for the treatment of default events, and default digital puts. Then we define the specification of the stochastic processes. We chose a multifactor Cox Ingersoll Ross (CIR) [4] square-root diffusion model both for the credit spread and for the default-free short rate. This setup can allow for arbitrary correlation between credit spreads and the risk-free term structure of interest rates. To keep the results as flexible as possible we used this specification only in the last step of the calculation of the prices, and we also give the expectations that have to be calculated if another model is used. This specification is used in the next section to price the credit risk derivatives that were introduced before. The pricing formulae are expressed in terms of the multivariate noncentral chi-squared distribution function, using results from Jamshidian [11]. The conclusion sums up the main results of the paper and points out the consequences that these results have for the further development of credit spread models.
2. Defaultable Bond Pricing with Cox Processes
The model used here is an extension of the Duffie-Singleton [7] model to multiple de- faults. More details to the model can be found in Sch¨onbucher [17, 18]. The new feature of this model is that a default does not lead to a liquidation but a reorganisation of the issuer: defaulted bonds lose a fraction q of their face value and continue to trade. This feature enables us to consider European-type payoffs in our derivatives without necessarily needing to specify a payoff of the derivative at default (although we will consider this case, too). All dynamics that we introduce are understood to be specified with respect to a common martingale measure. First we will have to introduce some notation:
2.0.1. The default-free term structure. The term structure of default-free bond prices is given by the time t prices P (t, T ) of the default-free zero coupon bonds with maturity T t. It is well-known that under the martingale measure ≥ R T r(s)ds (1) P (t, T ) = Et h e− t i , PRICING CREDIT RISK DERIVATIVES 3 where r denotes the default-free short rate. We will specify the dynamics of r later on. The discount factor βt,T over the interval [t, T ] is defined as R T r(s)ds (2) βt,T = e− t .
2.0.2. The defaultable bond prices. Defaultable bonds are the underlying securities to most credit risk derivatives. The time t price of a defaultable zero coupon bond with maturity T is denoted 1 with P 0(t, T ). The defaults themselves are modeled as follows: There is an increasing sequence of stopping times τi i IN that define the times of default. At each default the defaultable bond’s face{ value} ∈ is reduced by a factor q, where q may be a random variable itself. This means that a defaultable zero coupon bond’s final payoff is the product
Q(T ) := (1 qi)(3) Y − τi T ≤ of the face value reductions after all defaults until the maturity T of the defaultable bond. The loss quotas qi can be random variables drawn from a distribution K(dq) at time τi, but for the first calculations we will assume qi = q to be constant. To simplify notation the time of the first default will often be referred to with τ := τ1.
We assume that the times of default τi are generated by a Cox process. Intuitively, a Cox Process is defined as a Poisson process with stochastic intensity h (see Lando [13]). Formally the definition is: Definition 1. The default counting process
∞ (4) N(t) := max i τi t = 1 τi t X { ≤ } { | ≤ } i=1 is called a Cox process, if there is a nonnegative stochastic process h(t) (called the intensity of the Cox process), such that given the realization h(t) t>0 of the intensity, N(t) is a time-inhomogeneous Poisson process with intensity{ h(t)}.{ }
This means that the probability of having exactly n jumps, given the realisation of h, is n 1 T T (5) Pr N(T ) N(t) = n h(s) T s t = Z h(s)ds exp Z h(s)ds , − | { }{ ≥ ≥ } n! t {− t } and the probability of having n jumps (without knowledge of the realisation of h) is found by conditioning on the realisation of h within an outer expectation operator:
Pr [ N(T ) N(t) = n ] = Et Pr N(T ) N(t) = n h(s) T s t { ≥ ≥ } − − n | { } 1 T T (6) = Et Z h(s)ds exp Z h(s)ds , n! t {− t } 1Defaultable quantities usually carry a dash (’). 4 PHILIPP J. SCHONBUCHER¨
Specifically, the survival probability Ph(t, T ) of survival until time T as viewed from time t is R T h(s)ds (7) Ph(t, T ) = E 1 τ>T = E e− t . { } h i This application of the law of iterated expectations will prove extremely useful later on, it was first used in a credit risk context by Lando [13]. The specification of the default trigger process as a Cox process precludes a dependence of the default intensity on previous defaults and also ensures totally inaccessible stopping times τi as times of default. Apart from this it allows rich dynamics of the intensity process, specifically, we can reach stochastic credit spreads. Equations (5) and (6) will be used frequently later on. It is now easily seen (using the iterated expectations, see also Duffie and Singleton and Sch¨onbucher [7, 17, 18]) that in this setup the price of a defaultable zero coupon bond is given by R T r0(s)ds (8) P 0(t, T ) = Q(t)Et h e− t i .
The process r0 is called the defaultable short rate r0 and it is defined by
(9) r0 = r + hq. Here r is the default-free short rate, h the hazard rate of the defaults and q is the loss quota in default. If q is stochastic then q has to replaced by its (local) expectation e qt = qKt(dq) in equation (9). The similarity of equation (8) to the default-free bond pricingR equation (1) will be used later on.
It is convenient to decompose the defaultable bond price P 0 as follows:
(10) P 0(t, T ) = Q(t)P (t, T )Pˆ(t, T ). Here Q(t) denotes the number of defaults until now (time t). For all applications we will be able to set t = 0 and thus Q(t) = 1, but for the analysis at intermediate times 2 it is important to be clear about the notation . The defaultable bond price P 0(t, T ) is thus the product of Q(t), the influence of previous defaults, and the product of the default-free bond price P (t, T ) and the third factor Pˆ(t, T ) which is uniquely defined by equation (10), or equivalently: 1 P (t, T ) (11) Pˆ(t, T ) = 0 . Q(t) P (t, T ) Pˆ(t, T ) is related to the survival probability of the defaultable bond: If r and h are independent and there is a total loss q = 1 at default then Pˆ(t, T ) is the probability (under the martingale measure) that there is no default in [t, T ]. If r and h are not independent, Pˆ(t, T ) is the survival probability under the T -forward measure R T ˆ T t qh(s)ds (12) P (t, T ) = Et h e− i .
2For example, at the expiry date of an option we would like to separate previous defaults and credit spreads in the price of the underlying. PRICING CREDIT RISK DERIVATIVES 5 R R T T qh(s)ds T qh(s)ds (Under independence Et h e− t i and Et h e− t i coincide.)
3. The Credit Derivatives
This section contains a short overview of the specification and uses of the credit deriva- tives we will concentrate on later. The credit derivatives can be broadly classified into first generation products like default swaps and puts and default digital puts, and sec- ond generation products like Put options on defaultable bonds and Credit Spread Put options.
3.1. First Generation Products. The first generation CRDs that we will consider here are: default swap (also known as credit swap), default put, and default digital put. A default swap on a given defaultable bond (the reference security) is a swap contract where
counterparty A pays the swap rate s • until default or maturity T of the default swap, and conterparty B pays at default the difference between the recovery value of the • defaulted bond (measured by its post-default market price) and its notional value.
If there is no default of the reference security until T , counterparty B pays nothing. The reference security to a default swap is usually not a defaultable zero coupon bond but a defaultable coupon bond which trades close to par. Apart from the scarcity of defaultable zero coupon bonds in the markets this also takes care of another problem: the prepayment effect at early defaults. If there was an early default, a large fraction of counterparty B’s payment would not compensate for the default loss but it would compensate for the (risk-free) discounting of the final payment of the bond, whereas the default swap was only meant to cover the loss quota in default. The price of a default swap is closely related to the price of a defaultable floating rate note (FRN). Although the defaultable FRN is not a derivative, its pricing is nevertheless nontrivial because it does not always have to trade at par like default-free FRNs. A default put is similar to a default swap, with the difference, that counterparty A pays a lump sum payment up front instead of a regular rate until default. Again the same remarks with respect to the reference security of the default swap applies. The default digital put differs from the default put in that the payment of counterparty B at default is fixed in advance, i.e. B pays at default $ 1 to A. This mitigates potential problems in identifying the defaulted bond’s recovery value. For constant and known recovery rates the default digital put is equivalent to the default put. The default digital put is also an important building block of more complex credit derivatives which often imply a constant rebate at default. These derivatives can be viewed as derivatives that are knocked out at default plus a default digital put that pays off the rebate. 6 PHILIPP J. SCHONBUCHER¨ There is obviously also a swap version of the default digital put which is called a default digital swap. Here A pays the swap rate until default or until maturity of the contract. The valuation of a periodic payment until a fixed date in the future is standard and will not be treated here but we will consider the case when the fixed side of the swap is terminated at default and give the value of this payoff stream.
3.2. Second Generation Credit Risk Derivatives. The two types of second gener- ation instruments that we will discuss here are plain put options on defaultable bonds and credit spread put options. As with all credit risk derivatives it is very important how a default event is handled, i.e. whether the derivative covers the defaults or whether the contract is killed at default. Lump sum payments at default can be modelled as additional default digital put options, therefore we will not consider them here any more. A plain put option on a defaultable bond is different from classical bond options in that the price of the underlying (the defaultable bond) may jump downwards at defaults. It offers protection against rising default-free interest rates, rising credit spreads and (if the option survives defaults) against defaults. Concerning the treatment of defaults we will consider both, (a) the option is not modified at intermediate defaults and (b) the option is knocked out at default. A credit spread put option on a defaultable bond only conditions on the credit spread of the defaultable bond price over the spread of an otherwise identical default-free bond price. Credit spread puts can be viewed as exchange options, giving the bearer the right to exchange one defaultable bond for a certain number (< 1) of default free bonds. Again we can have survival and knockout at intermediate defaults. Obviously one can imagine similar contracts with American early-exercise features or with barriers on the spread etc., and with different underlying securities. These will have to be valued with a suitable numerical algorithm. Here we shall concentrate on the analytic valuation of the derivatives specified above.
4. Valuation of First Generation Credit Risk Derivatives
In this section we begin the discussion of the pricing of CRDs with the first-generation products that were introduced before. The simpler structure of these products allows us to keep our analysis at a more general level, we do not need to specify the stochastic differential equations satisfied by interest rates and credit spreads yet. The default digital put option and the default digital swap will be analysed first, as these products are the simplest and the pricing of the other products can often be reduced to the pricing of default digital puts. Next are defaultable FRNs and the default swap. The following data is needed: PRICING CREDIT RISK DERIVATIVES 7 the default-free term structure of bond prices R • T r(s)ds P (0,T ) = E h e− 0 i the defaultable bond prices under zero recovery R • T r(s)+h(s)ds P 0(0,T ) = E h e− 0 i the survival probabilities R • T h(s)ds Ph(0,T ) = E h e− 0 i.
We also need some information about the correlation between the default-free term structure of interest rates and the term structure of credit spreads. In this section we will assume independence of the two (which will yield Ph = P 0/P ), the case of correlation requires a more precise specification and will be treated in the following section. We assumed above (to simplify the notation) that the data is given for a recovery rate of zero (i.e. a loss quota q = 1 of one). This is especially easy to recover from the defaultable bond prices if the recovery is modeled in terms of equivalent default-free securities3 and the expected recovery rate is known:
In this model, 1 defaultable security F 0 becomes 1 c otherwise equivalent default-free securities F at default, where c can be stochastic.− Thus, the price of a defaultable zero coupon bond is given by
P 0(0,T ) = E β0,T 1 τ>T + cβ0,τ P (τ, T )1 τ T { } { ≤ } = E β0,T 1 τ>T + cE β0,T 1 τ T { } { ≤ } = E β0,T 1 τ>T + cE[ β0,T ] cE β0,T 1 τ>T { } − { } = (1 c)E β0,T 1 τ>T + cP (0,T ), − { } the defaultable zero coupon bond price can be decomposed into c default-free bonds and (1 c) defaultable bonds with zero recovery E β0,T 1 τ>T . Given information about c−, the prices of zero-recovery defaultable bonds are therefore{ } easily implied from P (0,T ) and P 0(0,T ). In the fractional recovery model it is not so easy to derive the value of a zero-recovery R T r(s)+h(s)ds defaultable bond E h e− 0 i just from knowledge of the recovery rate q, the R T r(s)+qh(s)ds defaultable bond price E h e− 0 i and the default-free bond prices. Never- theless, with a suitable choice of (time dependent) parameters both recovery models, fractional recovery (with q as loss quota) and recovery in terms of default-free securities (with c as default loss) can be transformed into each other: The value of the security in default is only expressed in different units, once in terms of defaultable bonds and once in terms of default-free bonds. Both approaches are therefore equivalent and one should use the specification that is best suited for the issue at hand. Here the recovery in terms of default-free securities has an advantage over the fractional recovery model.
3E.g. this model of recovery has been proposed by Lando [13] and Madan and Unal [16]. 8 PHILIPP J. SCHONBUCHER¨ The assumptions about the available data are open to criticism. One of the largest difficulties in credit risk modelling is the scarcity of useful data, and only rarely (maybe for some souvereign issuers) a full term structure of defaultable bond prices is available, let alone survival probabilities or recovery rates. These data problems will have to be addressed separately, here we just point out that all of the defaultable inputs (recovery rates, P 0(0,T ) or Ph(0,T )) can also stem from other sources than market prices, e.g. fundamental analysis or historical data of the same rating class or industry. The only requirement is that this information is given under risk-adjusted (pricing) probabilities, we are working under the martingale measure already. As shown above, the zero recovery assumption is mainly a tool to simplify the notation if defaultable bond prices, default-free bond prices and expected default losses c are known. It can be ignored if the survival probabilities are given directly. The assumption of independence on the other hand is much more restrictive. This assumption is needed to evaluate some expectations and can only be dropped in the full multifactor specification of interest-rate and spread dynamics in the following section.
4.1. Default Digital Payoffs. The default digital put option has the payoff 1 at the time of default. The timing is important because it determines until when the payoff has to be discounted, but we will consider a simpler specification in a first step: As a first example consider an European-style default digital put which pays off 1 at T iff there has been a default at some time before (or including) T . Its value is easily found (τ denotes the time of the first default): R T r(s)ds D =E e− 0 1 τ ˆ ˆ Pt =P (0, t)P (0, t)Et [ h(t)] =P (0, t)Pˆ(0, t)h(0, t) = P 0(0, t)h(0, t). R ˆ t h(s)ds Here P (0, t) = E h e− 0 i is the survival probability, and ∂ h(0, t) = ln Pˆ(0,T ) −∂T T =t is the associated ’forward rate’ of the spreads (for zero recovery q = 1). The survival probability Pˆ(0, t) was used as numeraire in the change of measure from the martingale ˆt t measure to P to remove exp 0 h(s)ds from the expectation. The final line follows from the well-known fact in{− risk-freeR interest} rate theory that under the T -forward measure the expectation of the short rate r(T ) at time T is exactly equal to the T - forward rate f(0,T ). Here we have a mathematically equivalent situation, just changing ˆ ˆ the notation such that P (0,T ) are our bond prices, PT is the forward measure, h(T ) is the short rate at T and h(0,T ) is the T -forward rate. Putting all together we gain the following representation for the value of a default digital put with maturity T under independence and zero recovery: T (17) D = Z P 0(0, t)h(0, t)dt. 0 The restrictive assumption of independence of interest rates and spreads will be relaxed later on. There is another simple representation of the default digital put if it is specified in a swap-like form. The swap specification is as follows: 10 PHILIPP J. SCHONBUCHER¨ Party A pays 1 at default if there is a default. Discounted this can be written as T β0,τ 1 τ T = Z β0,td1 τ T . { ≤ } 0 { ≤ } Party B pays a regular fee x(t)dt until default, i.e. he pays until maturity of the swap (also already in discounted form) T Z β0,tx(t)1 t τ dt 0 { ≤ } What should be the fair fee x(t)? If it can be chosen variable it should be the local credit spread h(t). The intuitive reason is that by Et 1 τ [t,t+dt] = h(t)dt { ∈ } h(t) it is the local default probability at time t. Suppose that the default protection is re-negotiated for every small time-interval [t, t + dt], then the fee should be h(t)dt for this interval. t Mathematically this follows from the fact that h(s)1 s τ ds is the predictable com- R0 { ≤ } pensator of 1 τ t , thus the expected values of stochastic integrals w.r.t. either of them are equal: { ≤ } T T Et [ β0,τ ] = Et Z β0,td1 τ t = Et Z β0,th(t)1 t τ dt . 0 { ≤ } 0 { ≤ } The expected discounted values of both sides of the swap are equal. This result is independent of the maturity of the swap and also valid for correlation between interest rates and defaults. The previous result required a continuous adaptation of the swap rate. If the swap rate x(t) is to be determined at the inception of the swap at t = 0 for all future times t T , then the value of the fixed leg is (again assuming the necessary regularity for the interchange≤ of integration and expectation) T E T β x(t)1 dt = x(t)E β 1 dt h 0 0,t τ t i Z 0,t τ t R { ≥ } 0 { ≥ } T (18) = Z x(t)P 0(0, t)dt. 0 Equation (18) gives the value of a constant or time-dependent income stream (i.e. a fixed swap leg) that is terminated at default or T . Equation (18) remains valid under correlation. The value of the fixed leg in equation (18) has to be set equal to the value of the variable (i.e. the default-protection) side of the swap, which is given (under independence) in equation (17): T Z P 0(0, t)h(0, t)dt. 0 PRICING CREDIT RISK DERIVATIVES 11 For a constant fixed rate x(t) = x this yields the fair swap rate for a default digital swap: T P 0(0, t)h(0, t)dt (19) x = R0 . T P (0, t)dt R0 0 Compare this to the plain vanilla fixed-for-floating swap rate in default-free interest rate model T 1 P (0,T ) 0 P (0, t)f(0, t)dt sr = − = R . T P (0, t)dt T P (0, t)dt R0 R0 ∂ (The equality follows from ∂t P (0, t) = f(0, t)P (0, t) and integrating by parts.) We see that the equation for the default digital− swap rate has a structure that is very similar to the structure of the equation for plain fixed-for-floating swap rates. The forward rate f(0, t) in the risk-free case has been replaced by the credit spread h(0, t) and the default-free bond prices P (0, t) are replaced by their defaultable counterparts P 0(0, t). In both cases the numerator denotes the value of the floating leg of the swap, and the denominator the fixed leg of the swap, where the floating leg for the default digital swap is the default contingent payment. The fact that the default swap is terminated at default introduces another difference to default-free swaps: The fixed leg has to be discounted using defaultable bond prices (P 0, the defaultable bond prices for zero recovery). If the swap was not killed at default, the fixed side would have to be discounted with default-free rates. 4.2. Defaultable FRNs and Default Swaps. There is a link between default digital puts and the prices of defaultable floating rate notes. Consider a defaultable floating rate note F 0(t, T ) with maturity T that pays a continuous coupon of r(t)dt until T and the principal of 1 at T . 4 Assume furthermore zero recovery in default. Then a portfolio of a defaultable floating rate note and a default digital put with the same maturity is fully hedged against default risk, its value is always one. Thus F 0(t, T ) = 1 D(t, T ), − where D(t, T ) is the value of the default digital put with maturity T . If there is a fractional recovery of (1 q) at default then this hedge becomes only − approximately valid: A portfolio of one F 0(t, T ) and q default digital puts D(t, T ) is protected against one default, but not against any more. Again, this can be resolved if the recovery is modeled in terms of equivalent default-free 5 securities : At default, 1 defaultable security F 0 becomes 1 c otherwise equivalent default-free securities F , where c can be stochastic. − 4Usually a defaultable FRN pays a coupon of r +s ¯ with a fixed spreads ¯ over the risk-free rate. We ignore these spread payments for now. They can be decomposed into a portfolio ofsdt ¯ zero coupon bonds of each maturity. 5See e.g. Lando [13] and Madan and Unal [16]. 12 PHILIPP J. SCHONBUCHER¨ For the floating rate notes this means that (if c is constant) they can be hedged with c default digital puts: F 0(t, T ) + cD(t, T ) = 1. If c is stochastic, this equality still holds for the prices under the martingale measure, but perfect hedging is obviously impossible, then. The pricing of the default swap can also be reduced to the pricing of a default digital put, it only differs from the default digital put in that the default swap pays off the loss quota c at default. Therefore it can be used to set up a perfect hedge of a defaultable FRN even for stochastic recovery rates. Instead of reiterating the analysis of the default digital put, we therefore use this property to give a characterisation of the default swap rate in terms of the credit spread of defaultable FRNs. Assume the following payoffs: The defaultable FRN pays the floating rate r(t) plus a constant spreads ¯ per time. The default swap rate is s, and the default-free FRN pays the floating rate r. Both floaters have a final payoff of 1. The payoffs are shown in table 1. defaultable FRN default swap default-free FRN Price at t = 0 F 0(0) 0 1 periodic payments (r(t) +s ¯)dt sdt r(t)dt − final payoff at t = T 1 + (r(t) +s ¯)dt sdt 1 + r(t)dt − value at default 1 c c 1 − Table 1. The Payoffs of defaultable FRN, default swap and default-free FRN. If the defaultable FRN trades at par at t = 0, i.e. F 0(0) = 1 then initially (at t = 0), at defaults and at t = T we have defaultable FRN + default swap = default-free FRN. As both the spreads ¯ of the defaultable FRN and the default swap rate s are constant they must coincide. Therefore s¯ = s, Find a defaultable FRN that trades at par and the spread s¯ of its coupons over r. Then s =s ¯ is the fair swap rate for a default swap of the same maturity. PRICING CREDIT RISK DERIVATIVES 13 This argument only uses a simple comparison of payoff schedules, it does not use any assumptions about the dynamics and distribution of risk-free interest rates or credit spreads or about the recovery rates. Finally we can use these results to price a defaultable FRN F 0 with a coupon of (r+s ¯)dt and principal of 1 (under zero recovery and independence): T T F 0(0,T ) = E r(t)β0,t1 τ>t dt + E sβ¯ 0,t1 τ>t dt + E β0,T 1 τ>T h R0 { } i h R0 { } i { } T T = Z E[ r(t)β0,t ]E 1 τ>t dt +s ¯Z P 0(0, t)dt + P 0(0,T ) 0 { } 0 T T ˆ = Z f(0, t)P (0, t)P (0, t)dt +s ¯Z P 0(0, t)dt + P 0(0,T ) 0 0 T = Z (f(0, t) +s ¯)P 0(0, t)dt + P 0(0,T ). 0 Again we see a strong similarity between the values of defaultable and default-free FRNs. The extension to positive recovery rates is not difficult now, and independence was only used in the evaluation of E r(t)β0,t1 τ>t . Setting F 0 = 1, solving fors ¯ and integrating by parts will transform this result{ into} the swap rate (19) of the default digital swap. 5. The Multifactor Model 5.1. Specification. In the previous section we could derive some pricing formulae for the first-generation CRDs without having to specify the dynamics we assume for credit spreads and interest rates, but we had to assume independence of the realisations of r and h in the final step of the analysis. For the pricing of the second generation credit risk derivatives and for correlation be- tween r and h we need a full specification of the dynamics of the risk-free interest rate r and the intensity of the default process h. A suitable specification should have the following properties: Both r and h are stochastic. Stochastic risk-free interest rates are a must anyway, • and a stochastic default intensity is required to reach a stochastic credit spread which is necessary for meaningful prices for the spread options. The dynamics of r and h are rich enough to allow for a realistic description of the • real-world prices. Duffie and Singleton [8] and Duffee [5] come to the conclusion that in many cases a multifactor model for the credit spreads is necessary. There should be scope to include correlation between credit spreads and risk-free • interest rates. Interest rates and credit spreads should remain positive at all times. Negative • credit spreads would represent an arbitrage opportunity but maybe relaxing this requirement in favour of a Gaussian specification is still acceptable because of the analytical tractability that is gained. 14 PHILIPP J. SCHONBUCHER¨ Here we chose a multifactor Cox Ingersoll Ross [4] setup, following mainly Jamshidian [11]. (Related models can also be found in Jamshidian [9, 10], Duffie and Kan [6], Chen and Scott [2] and Longstaff and Schwartz [15].) This model setup gives us the required properties while still retaining a large degree of analytical tractability. Furthermore, models of credit spreads of the CIR square-root type have been estimated by Duffie and Singleton [8] and Duffee [5]. Nevertheless, the derivation of the pricing formulae will be kept as general as possible to ensure that a transfer of the results to a different model specification (e.g. Gaussian) will be straightforward. Interest rates and default intensities are driven by n independent factors xi, i = 1, . . . , n with dynamics of the CIR square-root type: (20) dxi = (αi βixi)dt + σi√xidWi. − 1 2 We require αi > 2 σi to ensure strict positivity of the factors. The risk-free short rate is defined as a positive linear combination of these factors: n (21) r(t) = a x (t), X i i i=1 and the credit spread is another positive linear combination of these factors: n (22) h(t) = a x (t). X i0 i i=1 With all coefficients ai 0 and ai0 0 nonnegative we have ensured that r > 0 and h > 0 a.s. Unfortunately this≥ specification≥ can only generate positive correlation between r and h, if negative correlation is needed one could define modified factors xi0 that are negatively correlated to the xi by dxi0 = (αi βixi)dt σi√xidWi (note the minus in front of the Brownian motion). The analysis− would not− be changed significantly, only the changes of measure will be slightly different. Therefore we stick with the empirically more relevant case of positive correlation. Typically one would reserve the first m < n factors for the risk-free term structure (i.e. ai = 0 for i > m, m = 3 would usually be sufficient), and use the full set of state variables for the spreads (maybe n = 4 or n = 5, this will have to be determined empirically). The additional n m factors for the spreads ensure that the dynamics of the credit spreads have components− that are independent of the risk-free interest rate dynamics. The simplest example would be independence between r and h, where r = x1 and h = x2: one factor driving each rate, n = 2 m = 1 a1 = 1 a2 = 0 a10 = 0 a20 = 1. PRICING CREDIT RISK DERIVATIVES 15 In the general specification we have for a linear multiple cxi (c is a positive constant) of the factor xi (see CIR [4]) the following representation for a ‘bond price: T H2i(T t,c)cxi (23) Et exp Z cxi(s)ds = H1i(T t, c)e− − {− t } − where 2 1 2αi/σi (γ+βi)(T t) 2γe 2 − (24) H1i(T t, c) = γ(T t) − (γ + βi)(e 1) + 2γ − − γ(T t) 2(e − 1) (25) H2i(T t, c) = γ(T t)− − (γ + βi)(e 1) + 2γ − − 2 2 (26) γ = qβi + 2cσi . The default-free bond prices are given by R P R T T r(s)ds i aixi(s)ds P (t, T ) = Et h e− t i = Et h e− t i R R T a x (s)ds T a x (s)ds = E " e t i i # = E e t i i t Y − Y t h − i i i because the factors are independent. The bond price is thus a product of one-factor bond prices n (27) P (t, T ) = H (T, t, a )e H2i(T,t,ai)aixi(t). Y 1i i − i=1 Similarly, the defaultable bond prices are given by: n H (T,t,a +qa )(a +qa )x (28) P (t, T ) = Q(t) H (T, t, a + qa )e 2i i i0 i i0 i , 0 Y 1i i i0 − i=1 T because of P 0(t, T ) = Et exp r(s) + qh(s)ds . h {− Rt } i 5.2. Affine Combinations of independent non-central chi-squared distribu- tions. The mathematical tools for the analysis of the model have been provided by Jamshidian [11], who used them to price interest-rate derivatives in a risk-free interest rate environ- ment. Of these tools we need (a) the expressions for the evaluation of the expectations that will arise in the pricing equations, and (b) the methodology of a change of measure to remove discount factors from these expectations. First the distribution function of an affine combination of noncentral chi-squared random variables is presented. Most of the following expressions are based upon this distribution function. Then we consider the expressions for the expectations of noncentral chi- squared random variables, and finally the change-of-measure technique that has to be applied in this context. 16 PHILIPP J. SCHONBUCHER¨ 5.2.1. The distribution of noncentral chi-square random variables. The factors xi have square-root dynamics which give rise to noncentral chi-square dis- tributed final values6. The dynamics of a factor x under the martingale measure are (29) dx = (α βx)dt + σ√xdW. − The density function of the factor x(s) at time s given its value x(t) for an earlier time t < s is v q/2 (30) f(x(s), s; x(t), t) = ce u v I (2(uv)1/2), − − u q where Iq( ) is the modified Bessel function of the first kind of order q and · 2β 2α c = 2 β(s t) q = 2 1(31) σ (1 e− − ) σ − − β(s t) (32) u = cx(t)e− − v = cx(s). The distribution function of 2cx(s) given x(t) is χ2( 2cx(s) ; 2q + 2, 2u)(33) the noncentral chi-squared distribution function with 2q + 2 degrees of freedom and a parameter of noncentrality of 2u. As interest rates and spreads are affine combinations of these factors we need to analyse their distribution. Let Y be an affine function of n noncentral chi-square distributed random variables zi: n (34) Y = + η z , X i i i=1 where the zi are one-dimensional noncentral chi-squared distributed with 2 zi χ ( ; νi, λi) ∼ · νi degrees of freedom and noncentrality parameter λi. Call the distribution function of Y Y χ2( ; ν, λ, η, ), ∼ · where η, ν and λ are vectors, and for = 0 we will usually omit the last argument. The evaluation of this distribution function can be efficiently implemented via a fast Fourier transform of its characteristic function (see e.g. Chen and Scott (1992)). Al- ternatively, Jamshidian gives the following formula (which is basically the transform integral of the characteristic function): 2 1 1 ∞ ν 2 2 dξ (35) χ (y; ν, λ, η) = + Z Ψ( , λ, 2ξ η , 0) sin(ξy Φ(ν, λ, ξη)) , 2 π 0 2 − ξ 6See e.g. CIR [3, 4]. PRICING CREDIT RISK DERIVATIVES 17 where Y Ψ(ν, λ, η, ) = E e− is given by equation (39) and n νi γiλi Φ(ν, λ, γ) = arctan(2γi) + . X 2 1 + 4γ2 i=1 i The integral in (35) is numerically well-behaved in that the limit of the integrand as ξ 0 is finite and the integral is absolutely integrable. We will see later on that most valuation→ problems in the multifactor setup can be reduced to this one-dimensional integral. The evaluation of the noncentral chi-squared distribution function which one encounters in the one-factor independence case, also requires a numerical approximation, the implementation effort of the multifactor case therefore does not seem to be very much higher than the effort required for a model with one independent factor each for interest rates and spreads. 5.2.2. The values of some expectations. Let Y 0 be similarly distributed to Y with η0 and 0 respectively but the same zi, i.e. 2 2 (36) Y χ ( ; ν, λ, η, ),Y 0 χ ( ; ν, λ, η0, 0), ∼ · ∼ · and let y be a constant. Then the following expressions give: ... the expectation of Y : n (37) E[ Y ] = + η (λ + ν ), X i i i i=1 ... the distribution of Y : (38) Pr [ Y y ] = χ2(y ; ν, λ, η), ≤ − Y ... the expectation of e− : n Y 1 ηiλi (39) E e− = e− exp Y (1 + 2η )νi/2 1 + 2η i=1 i i Y y ... a call option on e− with strike e− : Y y + Y 2 λ η E (e− e− ) = E e− χ y ; ν, , − − 1 + 2η 1 + 2η y 2 e− χ (y ; ν, λ, η)(40) − − Y Y ... an exchange option on e− and e− 0 : Y Y + Y 2 λ η η0 E (e− e− 0 ) = E e− χ 0 ; ν, , − − − 1 + 2η 1 + 2η Y 2 λ η η0 (41) E e− 0 χ 0 ; ν, , − . − − 1 + 2η0 1 + 2η0 18 PHILIPP J. SCHONBUCHER¨ 5.3. The change of measure technique. The change of measure necessary to eliminate discounting with one of the factors is defined in the following Lemma which is due to Jamshidian [9]. The proof has been slightly adapted and can be found in the appendix: Lemma 1. Let x follow a CIR-type square-root process under the measure P : (42) dx = (α βx)dt + σ√x dW, − so that the model admits an affine term structure: R T x(s)ds H2x(T,t)x (43) G(x, t, T ) = Et h e− t i = H1x(T, t)e− . 1 2 and such that x = 0 is an unattainable boundary (α > 2 σ ). Then R T Z P t x(s)ds P Et h e− F (x(T )) i = G(x(t), t, T )Et [ F (x(T )) ](44) where under the measure P Z the process x has the dynamics 2 Z (45) dx = [α (β + H2x(T, t)σ )x]dt + σ√x dW , − with W Z a P Z -Brownian motion, and Z dW = dW H2x(T t)σ√xdt. − − The measure P Z is defined via the Radon-Nikodym density t Z(t) = Z σ(s) x(s)H2x(T, s)dW (s) . E − 0 p The dynamics of the other factors are unaffected. Remark: The lemma also holds for time-dependent parameters with α(t)/σ2(t) > 1/2, but here we only need the time-independent case. The distribution of x(T ) under the new measure P Z is given in the following lemma which is due to Jamshidian [9, 11]: Lemma 2. The time T -value x(T )/δT of the process x has a noncentral chi-squared distribution under P Z (i.e. given the x dynamics (45)), with ν degrees of freedom and noncentrality parameter λT x(t), where: 2 Pr [ x(T )/δT X ] = χ (X ; ν, λT x(t))(46) ≤ where σ2 δT = H2x(T t)(47) 4 − 4α (48) ν = σ2 ∂ 4 ∂T H2x(T t) (49) λT x(t) = 2 − x(t). σ H2x(T t) − PRICING CREDIT RISK DERIVATIVES 19 Proof: See Jamshidian [9, 11]. We will also need the same change of measure with a constant multiple y = cx of x. Lemmata 1 and 2 now take the following form: Lemma 3. The dynamics of x are given by (42) and The model admits an affine term structure: R T cx(s)ds H2x(T,t,c)cx (50) G(x, t, T, c) = Et h e− t i = H1x(T, t, c)e− . 1 2 and x = 0 is still an unattainable boundary (α > 2 σ ). Then R T P Z P t cx(s)ds c Et h e− F (x(T )) i = G(x(t), t, T, c)Et [ F (x(T )) ](51) Z where under the measure Pc the process x has the dynamics 2 Z (52) dx = [α (β + H2x(T, t)cσ )x]dt + σ√x dW , − Z Z with W a Pc -Brownian motion, and Z dW = dW H2x(T t, c)cσ√xdt. − − Z The measure Pc is defined via the Radon-Nikodym density t Z(t) = Z σ(s)c x(s)H2x(T, s, c)dW (s) . E − 0 p And lemma 2 becomes now ˆ Lemma 4. The time T -value x(T )/δT of the process x has a noncentral chi-squared Z distribution under Pc (i.e. given the x dynamics (52)), with νˆ degrees of freedom and ˆ noncentrality parameter λT x(t), where: ˆ 2 ˆ Pr x(T )/δT X = χ (X ;ν, ˆ λT x(t))(53) h ≤ i where 2 ˆ σ δT = H2x(T t, c)(54) 4 − 4α (55) νˆ = σ2 ∂ H (T t, c) ˆ 4 ∂T 2x (56) λT x(t) = 2 − x(t). cσ H2x(T t, c) − Proof: Apply lemmata 1 and 2 to y = cx. Note that y has the dynamics (57) dy = d(cx) = (cα βy)dt + σ√c √y dW − (58) =: (ˆα βyˆ )dt +σ ˆ√y dW, − which is still of the CIR square-root form. R Z T cx(t)dt The change of measure to Pc removes the discounting with e− 0 , while changing ˆ the distribution of δT x(T ) to a noncentral chi-squared distribution with parameters given in equations (54) to (56). 20 PHILIPP J. SCHONBUCHER¨ 6. Valuation of Second Generation Credit Risk Derivatives 6.1. Credit Spread Put. A Credit Spread Put on a defaultable bond P 0(t, T2) with maturity T1 < T2 gives the holder the right to sell defaultable bonds at time T1 at a price that corresponds to a yield spread ofs ¯ above the yield of an (otherwise identical) ¯ s¯(T2 T1) default-free bond P (T1,T2). Define the exchange ratio S := e− − . These credit ¯ derivatives entitle the holder to exchange one defaultable bond P 0(T1,T2) for S default- free bonds P (T1,T2) at time T1. The payoff function is: ¯ + (59) (SP (T1,T2) P 0(T1,T2)) − Here the default is a jump in the price of the underlying security, we will first consider the case when the contract is not modified at default, then the case when the option is knocked out by a default. In the first case the payoff is conditioned on the full defaultable bond price P 0(T1,T2), and the default risk is borne by the writer of the security. The holder of the security is protected by the contract against any losses worse than a final credit spread ofs ¯ and against all defaults, but not against interest rate risk. The exchange option is therefore well suited to lay off most of the credit exposure in the underlying instrument while keeping some limited exposure to the credit spread of the underlying. For the valuation of the exchange option we find ¯ + Dt = Et βt,T1 (SP (T1,T2) P 0(T1,T2)) R − T1 t r(s)+h(s)ds ¯ ˆ + = Et e− P (T1,T2)(S P (T1,T2)) h − i ¯ + SP (t, T2) P 0(t, T2) R − T1 t r(s)+h(s)ds ¯ ˆ (60) Et e− P (T1,T2)(S P (T1,T2)) . − h − i This is reached as follows: The expectation to evaluate is (assuming no defaults so far Q(t) = 1) ¯ + Dt = Et βt,T (SP (T1,T2) P 0(T1,T2)) 1 − ¯ ˆ + = Et βt,T (SP (T1,T2) P (T1,T2)Q(T1)P (T1,T2)) h 1 − i ¯ ˆ + = Et βt,T P (T1,T2)(S Q(T1)P (T1,T2)) h 1 − i ¯ ˆ + = Et 1 τ>T1 βt,T1 P (T1,T2)(S P (T1,T2)) h { } − i ¯ ˆ + + Et 1 τ T1 βt,T1 P (T1,T2)(S Q(T1)P (T1,T2)) . h { ≤ } − i where we split up the payoff in the case without intermediate default (1 τ>T1 ) and the { } case with intermediate default (1 τ T1 ). If there has been an intermediate default, we assume that the option is in the money{ ≤ } and we omit the maximum sign for that event: ¯ ˆ + = Et 1 τ>T1 βt,T1 P (T1,T2)(S P (T1,T2)) h { } − i PRICING CREDIT RISK DERIVATIVES 21 ¯ ˆ + Et 1 τ T1 βt,T1 P (T1,T2)(S Q(T1)P (T1,T2)) , h { ≤ } − i and then substitute 1 τ T1 = 1 1 τ>T1 to reach { ≤ } − { } ¯ ˆ + = Et 1 τ>T1 βt,T1 P (T1,T2)(S P (T1,T2)) h { } − i ¯ ˆ + Et βt,T P (T1,T2)(S Q(T1)P (T1,T2)) h 1 − i ¯ ˆ Et 1 τ>T1 βt,T1 P (T1,T2)(S Q(T1)P (T1,T2)) − h { } − i ¯ ˆ + = Et 1 τ>T1 βt,T1 P (T1,T2)(S P (T1,T2)) h { } − i ¯ ˆ + Et βt,T P (T1,T2)S Et βt,T Q(T1)P (T1,T2)) 1 − h 1 i ¯ ˆ Et 1 τ>T1 βt,T1 P (T1,T2)(S Q(T1)P (T1,T2)) − h { } − i ¯ ˆ + = Et 1 τ>T1 βt,T1 P (T1,T2)(S P (T1,T2)) h { } − i ¯ + SP (t, T2) P 0(t, T2)) − ¯ ˆ Et 1 τ>T1 βt,T1 P (T1,T2)(S P (T1,T2)) . − h { } − i In the last line we used the fact that 1 τ>T1 Q(T1) = 1 τ>T1 , the indicator function of { } { } ’no default until T1’ switches off the ’influences Q(T1) of default until T1’. From the decomposition of the payoff in a ’no default’ part and a ’default part’ we can also directly see the expectation for the price of a credit spread option that is killed at default: It is ˜ ¯ + (61) Dt = Et βt,T (SP (T1,T2) P 0(T1,T2)) . 1 − Next we have to eliminate the indicator functions from the expectations. This is done by conditioning on the realisation of h and using the Cox process properties of the default triggering process: ¯ ˆ + Et 1 τ>T1 βt,T1 P (T1,T2)(S P (T1,T2)) h { } − i ¯ ˆ + = Et E 1 τ>T1 βt,T1 P (T1,T2)(S P (T1,T2)) h h h { } i i R − T1 t h(s)ds ¯ ˆ + = Et e− βt,T P (T1,T2)(S P (T1,T2)) . h 1 − i The second expectation is treated similarly to reach equation (60). Until now we have not used the multifactor CIR specification yet, the evaluation of the expectations (60) and (61) could be implemented with any other specification, too. To eliminate the integrals with the exponentials of r + h ds we will have to perform a change of measure for each factor involved. RecallingR that T1 n T1 r(s) + h(s)ds = (a + a )x (s)ds Z X Z i i0 i − t − i=1 t 22 PHILIPP J. SCHONBUCHER¨ we have to eliminate each factor individually, using Lemma 1. Because all factors are independent the change of measure procedure is not difficult and follows directly: For every function F (x(T1)) of the factors at T1 we have R T1 t r(s)+h(s)ds (62) Et h e− F (x(T1)) i n rh = EP [ F (X(T )] H (T , t, a + a ) exp H (T , t, a + a )(a + a )x (t) , t 1 Y 1 1 i i0 2 1 i i0 i i0 i i=1 {− } rh rh where under the measure P the factor xi(T )/δi is noncentral chi-square distributed rh rh with νi degrees of freedom and parameter of noncentrality λi : 2 rh (ai + ai0 )σi δ = H2(T1 t, ai + a0 )(63) i 4 − i rh 4αi (64) νi = 2 σi rh xi(t) ∂ (65) λi = rh H2(T1 t, ai + ai0 ). δi ∂T − To evaluate the remaining expectations we define the following random variables: ¯ Y := ln(SP (T1,T2)) n n rh xi (66) =¯s(T2 T1) + ln H1i(T2 T1, ai) δ H2i(T2 T1, ai) X X i δrh − i=1 − − i=1 − i ˆ Y 0 := ln(P (T1,T2)P (T1,T2)) n n rh xi (67) = ln H1i(T2 T1, ai + qa0 ) δ H2i(T2 T1, ai + qa0 ) . X i X i i δrh i=1 − − i=1 − i Note that both Y and Y 0 have a representation as an linear affine combination of rh rh (under P ) noncentral chi-square distributed random variables xi(T1)/δi . Therefore their distributions (using the notation of Jamshidian [11]) are n (68) Y χ2 νrh; λrh; δrhH (T T , a);s ¯(T T ) + ln H (T T ; a )! 2 2 1 2 1 X 1i 2 1 i ≡ − − − i=1 − n (69) Y χ2 νrh; λrh; δrhH (T T , a + qa ); ln H (T T ; a + qa )! 0 2 2 1 0 X 1i 2 1 i i0 ≡ − − i=1 − rh rh P Y P Y 0 The closed form solutions for the expectations Et e ,Et e and for the rh P Y Y 0 + option Et (e e ) can be read from equations (37) to (41). Substitution of these solutions and− (62) into (60) yields the closed-form solution for the price of the exchange option. 6.2. Put Options on Defaultable Bonds. The credit risk derivatives considered so far conditioned on the spread of the defaultable bond over a default-free bond, but the reference to a default-free bond is not necessary, we can also consider derivatives on the PRICING CREDIT RISK DERIVATIVES 23 defaultable bond alone, e.g. a Put or call on a defaultable bond. Again we can consider the question whether the Put covers default losses, or whether it is killed at default. A European Put with maturity T1 and strike K on a defaultable bond P 0(t, T2) pays off + (70) (K P 0(T1,T2)) . − This derivative protects the buyer against all risks: Interest rates, credit spreads or defaults that could move the defaultable bond price below K. To price this derivative we have to evaluate R T1 r(s)ds + Et e− t (K P 0(T1,T2)) . h − i This can be expanded by the now familiar technique of first conditioning on the reali- sation of h and then substituting the event probabilities: R T1 t r(s)ds + Et h e− (K P 0(T1,T2)) i R − T1 t r(s)+h(s)ds ¯ + = Et e− (K P (T1,T2)) h − i + P (t, T1)K P 0(t, T2) R − T1 t r(s)+h(s)ds ¯ (71) Et e− (K P (T1,T2)) . − h − i Equation (71) is very similar to the pricing equation (60) of the exchange option. The ¯ only difference is that SP (T1,T2) in (60) has been replaced with K in (71). The valuation is therefore very similar: The same change of measure removes the discounting, the only difference is that equation (39) (the plain option) instead of (40) (the exchange option) is to be used in the final substitution. It is important to note that the value of this option is not the value of an European Put on a bond in a world where the short rate is r0 = r + qh. (One might be lead to this view by the simple representation (8) of the defaultable bond prices.) This view would ignore the large jumps the bond price makes at defaults and only price the spread component. To illustrate this point we introduce a new security: the put option on the yield with security protection. This option is the one that is similar to a default-free put option in a world with interest rate r0 = r + qh, but its specification conditions on the default model that is used, and therefore will not be viable in real-world applications. 6.2.1. Put option on the yield of a defaultable bond: Security protection. A European Put with security protection, maturity T1 and strike K on the yield of a defaultable bond P 0(t, T2) pays off ¯ + (72) Q(T1)(K P (T1,T2)) . − The term ’security protection’ has been chosen to point out the special feature in this specification: At each default the face value of the defaultable bond is reduced by a factor 1 q. This payoff function is also reduced by the same factor, such that this option always− covers exactly one defaultable bond, the size of the protection is adjusted 24 PHILIPP J. SCHONBUCHER¨ to the ’size’ (=face value) of the security. Therefore we called it Put with security protection. This security can be used to artificially shorten the maturity of a defaultable bond P 0(t, T2): With K = 1 a portfolio of this credit risk derivative and the underlying defaultable bond has the same payoff as a defaultable bond with maturity T1: the payoff is ¯ ¯ P 0(T1,T2) + Q(T1)(1 P (T1,T2)) = P 0(T1,T2) + Q(T1) Q(T1)P (T1,T2)) = Q(T1). − − Again very similarly to (60) we reach the expectation R T1 t r(s)+qh(s)ds ¯ + Et e− (K P (T1,T2)) , h − i which is exactly identical to the expectation that has to be evaluated to price a put option on a zero coupon bond in a world where r0 = r + qh is the riskless short rate. To evaluate the expectation in the multifactor CIR-framework we have to perform the same change of measure as for the credit spread put with security protection, again only ¯ SP (T1,T2) has been replaced by K. 7. Conclusion In this paper we demonstrated the Cox process approach to the pricing of credit risk derivatives, and some closed form solutions were given for the case of a multifactor CIR-specification of the credit spread and interest rate dynamics. The Cox process modelling approach derives its tractability from the fact that the pricing of derivatives can be done in stages: First, by conditioning on the realisation of the intensity process h the pricing problem can be reduced to the fairly straightforward case of inhomogeneous Poisson processes, for which most expected values are easily calculated. Thus the explicit reference to default events can be eliminated and replaced by expressions in the intensity of the default process. After this stage the methods of risk-free interest rate theory can be applied, because the reference to jump processes has disappeared. We used specifically the change of measure technique in several to remove discount T factors of the form exp 0 g(s)ds where g is an ’interest-rate like’ process, and then some well-known properties− R of the final distributions of the factor processes. In most cases we needed to change the measure to remove the discount factor R T r(t)+h(t)dt e− 0 which typically arose from expectations of the form R R T r(t)dt T r(t)+h(t)dt E e− 0 1 τ>T X = E e− 0 X , h { } i h i i.e. survival contingent payoffs. The resulting measure can be termed a default measure. The change of measure after the elimination of the default indicator function makes sure that the new measure remains equivalent to the original martingale measure. Simply PRICING CREDIT RISK DERIVATIVES 25 taking a defaultable bond as numeraire would mean that this property is lost, and the change of measure would become much more complicated because the price path of a defaultable bond can be discontinuous. Even when no closed form solutions are available the conditioning technique is still very useful as pre-processing procedure for a subsequent numerical solution of the pricing equation. Especially Monte Carlo methods are notoriously slow to converge for low- probability events (like defaults), the results can be speeded up significantly if the default events have been removed. P.d.e. methods (like finite differences solvers) are usually based on a Feynman-Kac representation of the price-expectation as an expectation of a stochastic integral and a final payoff in terms of diffusion processes. Therefore they cannot handle discrete default events directly7, a pre-processing to remove the jumps is necessary here, too. The only case when this does not make sense is if one would like to recover the full distribution of the payoffs from the numerical scheme, including default events. It was seen furthermore that the analogy to the risk-free interest rate world does not carry through completely. There is the simple representation (8) of a defaultable bond price as ’bond price expectation’ with defaultable interest rates given by r0 = r + qh but this does not mean that options on defaultable bonds can also be treated as if there were no defaults but just a new interest rate r0. In the last section this was shown to be quite wrong. Most of the results of this paper are independent of the specification used for the interest- rate and credit spread processes, the point of the paper was to present the techniques that have to be used in all specifications and the transfer to another specification should not be too hard. Another point to mention is the amount of data and information that is needed to price credit risk derivatives. One of the advantages of the Duffie-Singelton setup appeared to be the fact, that default intensities h and default loss rates q only appeared linked as a product in the bond pricing equation, an qh could therefore be estimated from the credit spreads – without any further need for separate default and recovery information. This breaks down when the credit risk derivatives are considered, the default intensity h does appear separate from the default losses q, both pieces of information are needed separately. In a practical implementation much of this data will not be available and the financial engineer will have to resort to historical data, market data from other issuers of the same industry, region and rating class, or to numbers from fundamental analysis. On the other hand, every derivative needs an underlying, and this underlying should be fairly immune to market manipulation. This means that second generation products ar only viable iff there is some degree of liquidity in the market for the underlying defaultable bond. Liquidity can usually only be expected for large issuers and this means better quality on historical price data on the one hand and probably the existence of further 7It is possible to incorporate the jump processes in a finite-difference scheme, but at the cost of computing speed and having full matrices. 26 PHILIPP J. SCHONBUCHER¨ defaultable bonds by the same issuer that trade in liquid markets and that can be used to build a credit spread curve. Therefore CRDs will only be viable in markets and for issuers where the data problems are less severe anyway. Nevertheless, data availability and quality is the most pressing problem in the field at the moment. PRICING CREDIT RISK DERIVATIVES 27 Appendix A. Appendix Proof of Lemma 1. The proof of Lemma 1 is an application of the Girsanov theorem and the change of measure technique: Define R t x(s)ds (73) g(0, t) = e− 0 . Then 1 Z(t) := g(0, t)G(x(t), t, T )(74) G(x(0), 0,T ) is a positive martingale with initial value Z(0) = 1 and can therefore be used as a Radon- Nikodym density for a change of measure from P to P Z defined by dP/dP Z = Z, and t Z(t) = Z σ x(s)H2x(T s)dW (s) . E − 0 p − Then R Z T P P P t x(s)ds Z(t)Et [ F (x(T )) ] = Et [ Z(T )F (x(T )) ] = Et h e− F (x(T )) i It remains to show that x has indeed the postulated dynamics under P Z . The P dynamics of x are dx = κ(α x)dt + σ√x dW − where W is a P -Brownian motion. By Girsanov’s theorem Z dW = dW H2x(T t)σ√xdt − − where W Z is a P Z -Brownian motion. Thus 2 Z dx = κ(α x)dt H2x(T t)σ xdt + σ√x dW − − − 2 Z (75) = [κα σ (κ + H2x(T t))x]dt + σ√x dW . − − The invariance of the dynamics of the other factors follows from the independence of the individual factors. 28 PHILIPP J. SCHONBUCHER¨ References [1] P. Artzner and F. Delbaen, Credit risk and prepayment option, Astin Bulletin, 22 (1992), pp. 81–96. [2] R.-R. Chen and L. Scott, Interest rate options in multifactor cox-ingersoll-ross models of the term structure, The Journal of Derivatives, 3 (1995), pp. 52–72. [3] J. Cox, J. E. Ingersoll, and S. A. Ross, An intertemporal general equilibrium model of asset prices, Econometrica, 53 (1985), pp. 363–384. [4] , A theory of the term structure of interest rates, Econometrica, 53 (1985), pp. 385–407. [5] G. R. Duffee, Estimating the price of default risk, working paper, Federal Reserve Board, Wash- ington DC, September 1995. [6] D. Duffie and R. Kan, A yield-factor model of interest rates, Mathematical Finance, 6 (1996), pp. 379–406. [7] D. Duffie and K. Singleton, Modeling term structures of defaultable bonds, working paper, The Graduate School of Business, Stanford University, Stanford, June 1994. revised June 1996. [8] D. Duffie and K. Singleton, An econometric model of the term structure of interest rate swap yields, Journal of Finance, 52 (1997), pp. 1287–1321. [9] F. Jamshidian, Pricing of contingent claims in the one-factor term structure model, working paper, Sakura Global Capital, July 1987. [10] , A simple class of square-root interest-rate models, Applied Mathematical Finance, 2 (1995), pp. 61–72. [11] , Bond futures and option evaluation in the quadratic interest rate model, Applied Mathe- matical Finance, 3 (1996), pp. 93–115. [12] R. A. Jarrow and S. M. Turnbull, Pricing derivatives on financial securities subject to credit risk, Journal of Finance, 50 (1995), pp. 53–85. [13] D. Lando, On Cox processes and credit risky bonds, working paper, Institute of Mathematical Statistics, University of Copenhagen, March 1994. revised December 1994. [14] , Three Essays on Contingent Claims Pricing, PhD thesis, Graduate School of Management, Cornell University, 1994. [15] F. Longstaff and E. Schwartz, Interest rate volatility and the term structure: A two factor general equilibrium model, The Journal of Finance, 47 (1992), pp. 1259–82. [16] D. B. Madan and H. Unal, Pricing the risks of default, working paper, College of Business and Management, University of Maryland, March 1994. [17] P. J. Schonbucher¨ , The term structure of defaultable bond prices, Discussion Paper B-384, University of Bonn, SFB 303, August 1996. [18] , Term structure modelling of defaultable bonds, The Review of Derivatives Studies, Special Issue: Credit Risk, (1998). Department of Statistics, Faculty of Economics, University of Bonn,, Adenauerallee 24-42, 53113 Bonn, Germany E-mail address: [email protected]